Is it possible to approximate some PDE semigroups by explicit methods?

Question

I'm concerned with the numerical methods for the approximations of semigroup associated to following Cauchy problems (which typically involves unbounded operators):

$\begin{equation} \begin{array}{ccc} \frac{du}{dt} + Au& = & 0 \\ u(0) &=& u_0. \end{array} \end{equation}$ (1)

When $A$ is an m-accretive operator, it is known [1] that, a solution $u(t)$ can be given by an exponential formula:

$ u(t) = \lim\limits_{n \rightarrow \infty} [(I + \frac{t}{n} A)^{-1}]^n u_0, $ (2)

which can be seen as an implicit Euler discretization scheme in time for the Cauchy problem (1).

Question :

Is it possible for some operators A (typically a laplacian) to approximate the semigroup using an explicit scheme such as :

$U_{k+1} = U_k + hAU_k.$ (3)

Ref: [1] Haïm Brezis "Analyse fonctionnelle et applications"

Answer 1

You can find some useful information in the manuscript:

P. Csomós, A. Bátkai, B. Farkas and A. Ostermann, Operator Semigroups for Numerical Analysis 2013.